The proof follows from the general properties of involutions (see InvolutiveHomography.html ). It suffices to show that F, defined by the previous recipe is an homography. But this is seen at once by taking a

The following arguments identify this involution with one which constructs explicitly P and its polar (line e=OD

[1] Consider first the tangent line and the normal of the conic at D. Let O be the intersection-point of the tangents at D and D

[2] From D draw a parallel to the tangent [OD

[3] Let L be the intersection point of lines [OD

[4] Points P, D

[5] It follows that for every point C on line OD

[6] From its definition D

[7] To solve this exercise parametrize the bundle through a line orthogonal to the bisector of the two lines at right angle, with origin at the intersection point with the bisector. In this system the parameters of the lines in involution satisfy an equation of the form Axx'+B(x+x')+C=0. For the orthogonal lines we have x'=-x=r (constant) hence C-r

See FregierPolar.html for another easy way to construct point P and its corresponding polar line.

GoodParametrization.html

InvolutiveHomography.html

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